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Bogatkov and Babadagli
2010
and Fourar and Radilla
2009
). This behaviour is
characterized by an early tracer breakthrough and long-decaying tails. It has been
found that classical advection-dispersion equations do not properly describe the tracer
breakthrough behaviour at long times as other approaches do, particularly fractional
advection-dispersion models. Therefore, parameter estimation based on anomalous
transport models becomes very relevant (Suzuki et al.
2012
; Chakraborty et al.
2009
).
Among other approaches for studying anomalous transport, that do not have the
high computational costs involved in evaluating fractional time derivatives, are the
fractal continuum models (Herrera-Hernández et al.
2013
). In this paper, we present
a methodology for parameter estimation for one of the models described in this
previously mentioned work. It is a one-dimensional advective-dispersive model in
which the space is a fractal continuum (Tarasov
2005
) and dispersion follows a
power-law dependence with the length scale (Sahimi
1993
).
The paper is planned as follows: In Sect.
2
the mathematical model and its solution
are presented. The procedure for parameter estimation is described in Sect.
3
. Further,
in Sect.
4
the generation of synthetic tracer breakthrough data is discussed, and in
Sect.
5
the robustness of the estimation method is examined. Finally, the results are
analysed in Sect.
6
and conclusions are given in Sect.
7
.
2 The Model
2.1 The Fractal Continuum Model
The systemwe consider consist of an underground fractured porous formation where
uniform flow takes place. Fluid and tracer are injected in a plate (located at
x
=
x
w
)
and extracted at another plate (at
x
L
) as illustrated in Fig.
1
, where the planes
are perpendicular to the paper sheet. This reproduces common assumptions made in
Euclidean inter-well one-dimensional flow models (Bear
1972
; Chaberneau
2000
).
We consider a porous medium that is Euclidean in the
y
-
z
plane and fractal in
x
direction. Here a Cartesian coordinate system with the
x
-axes perpendicular to the
plates is employed. The fractality is introduced as a fractal continuumand a geometric
mapping of the fractal space into the Euclidean space is proposed. The fractal element
of volume is written in terms of fractal and Euclidean elements as in Balankin
2012
;
Hernandez-Coronado et al.
2012
; Herrera-Hernández et al.
2013
=
dV
α
=
d
α
dS
2
,
(1)
where
dS
2
is the Euclidean cross-section element and
d
α
is the fractal length ele-
ment, whose definition according to Ostoja-Starzewski (
2009
)is
ˆ
ʓ(α)
|
x
0
|
α
−
1
d
α
=
c
1
(
x
;
α,
x
0
)
dx
,
c
1
(
x
;
α,
x
0
)
=
x
−
,
(2)
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